Let $\mathcal{P}$ be the parabola in the plane determined by the equation $y = x^2.$  Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points.  If three of these points are $(-28,784),$ $(-2,4),$ and $(13,169),$ find the sum of the distances from the focus of $\mathcal{P}$ to all four of the intersection points.
Explanation: Let the four intersection points be $(a,a^2),$ $(b,b^2),$ $(c,c^2),$ and $(d,d^2).$  Let the equation of the circle be
\[(x - k)^2 + (y - h)^2 = r^2.\]Substituting $y = x^2,$ we get
\[(x - k)^2 + (x^2 - h)^2 = r^2.\]Expanding this equation, we get a fourth degree polynomial whose roots are $a,$ $b,$ $c,$ and $d.$  Furthermore, the coefficient of $x^3$ is 0, so by Vieta's formulas, $a + b + c + d = 0.$

We are given that three intersection points are $(-28,784),$ $(-2,4),$ and $(13,196),$ so the fourth root is $-((-28) + (-2) + 13) = 17.$

The distance from the focus to a point on the parabola is equal to the distance from the point to the directrix, which is $y = -\frac{1}{4}.$  Thus, the sum of the distances is
\[784 + \frac{1}{4} + 4 + \frac{1}{4} + 169 + \frac{1}{4} + 17^2 + \frac{1}{4} = \boxed{1247}.\]